For each model $\mathcal{X}$, the subgroup of $C^{0}(X)$ spanned $\log|\mathfrak{a}|$ with $\mathfrak{a}$ ranging over all vertical (fractional) ideal sheaves of $\mathcal{X}$ coincides with $\mathcal{D}(X)_{\mathbf{Z}}$. It is furthermore stable under max and separates points of $X$.